Note on the Thomas–Fermi and Thomas–Fermi–Dirac model calculation of atomic polarizabilities

Abstract

This paper discusses a recent calculation by Bruch and Lehnen of the static electric dipole polarizabilities of neutral inert gas atoms. These authors obtained the polarizabilities within the framework of statistical models by making use of an energy functional method. The method, as applied by Bruch and Lehnen, makes use of the exact solutions of either the Thomas–Fermi (TF) or of the Thomas–Fermi–Dirac (TFD) equations. This paper points out that a better agreement than the one obtained by Bruch and Lehnen between calculated and experimental values of the atomic polarizabilities can be obtained upon resorting to approximate (analytical) variational solutions of the TF and TFD equations. The improved agreement, in the case of the TF model, is attributed to the fact that the variational solution of the TF equation permits one to construct a radial electron density for a neutral atom that decreases with the distance from the nucleus exponentially. That this is an improvement is seen from the fact that in the exact TF model the radial electron density drops off with the distance from the nucleus as the inverse fourth power of this quantity. Essentially the same situation prevails in the case of the approximate (analytical) solution of the TFD equation, which makes use of the variational solution for a TF atom. In this case, the improved radial electron density manifests itself in a contraction of the atomic radius, relative to that which is obtained within the exact TFD model.